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Continuity equation
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==General equation== ===Definition of flux=== {{main|Flux}} A continuity equation is useful when a '''flux''' can be defined. To define flux, first there must be a quantity {{math|''q''}} which can flow or move, such as [[mass]], [[energy]], [[electric charge]], [[momentum]], number of molecules, etc. Let {{math|''ρ''}} be the volume [[density]] of this quantity, that is, the amount of {{math|''q''}} per unit volume. The way that this quantity {{math|''q''}} is flowing is described by its flux. The flux of {{math|''q''}} is a [[vector field]], which we denote as '''j'''. Here are some examples and properties of flux: * The dimension of flux is "amount of {{math|''q''}} flowing per unit time, through a unit area". For example, in the mass continuity equation for flowing water, if 1 gram per second of water is flowing through a pipe with cross-sectional area 1 cm<sup>2</sup>, then the average mass flux {{math|'''j'''}} inside the pipe is {{nowrap|(1 g/s) / cm<sup>2</sup>}}, and its direction is along the pipe in the direction that the water is flowing. Outside the pipe, where there is no water, the flux is zero. * If there is a [[velocity field]] {{math|'''u'''}} which describes the relevant flow—in other words, if all of the quantity {{math|''q''}} at a point {{math|'''x'''}} is moving with velocity {{math|'''u'''('''x''')}}—then the flux is by definition equal to the density times the velocity field: : <math display="block">\mathbf{j} = \rho \mathbf{u}</math> : For example, if in the mass continuity equation for flowing water, {{math|'''u'''}} is the water's velocity at each point, and {{math|''ρ''}} is the water's density at each point, then {{math|'''j'''}} would be the mass flux, also known as the material [[Discharge (hydrology)|discharge]]. * In a well-known example, the flux of [[electric charge]] is the [[electric current density]]. [[File:Continuity eqn open surface.svg|390px|right|thumb|Illustration of how the fluxes, or flux densities, {{math|'''j'''<sub>1</sub>}} and {{math|'''j'''<sub>2</sub>}} of a quantity {{math|''q''}} pass through open surfaces {{math|''S''<sub>1</sub>}} and {{math|''S''<sub>2</sub>}}. (vectors {{math|'''S'''<sub>1</sub>}} and {{math|'''S'''<sub>2</sub>}} represent [[vector area]]s that can be differentiated into infinitesimal area elements).]] * If there is an imaginary surface {{math|''S''}}, then the [[surface integral]] of flux over {{math|''S''}} is equal to the amount of {{math|''q''}} that is passing through the surface {{math|''S''}} per unit time: {{Equation box 1 |indent=: |equation = <math> (\text{Rate that }q\text{ is flowing through the imaginary surface }S) = \iint_S \mathbf{j} \cdot d\mathbf{S}</math> |cellpadding |border |border colour = #50C878 |background colour = #ECFCF4 }} : in which <math display="inline">\iint_S d\mathbf{S}</math> is a [[surface integral]]. (Note that the concept that is here called "flux" is alternatively termed '''flux density''' in some literature, in which context "flux" denotes the surface integral of flux density. See the main article on [[Flux]] for details.) ==={{anchor|Integral form|integral form}} Integral form=== The integral form of the continuity equation states that: * The amount of {{math|''q''}} in a region increases when additional {{math|''q''}} flows inward through the surface of the region, and decreases when it flows outward; * The amount of {{math|''q''}} in a region increases when new {{math|''q''}} is created inside the region, and decreases when {{math|''q''}} is destroyed; * Apart from these two processes, there is ''no other way'' for the amount of {{math|''q''}} in a region to change. Mathematically, the integral form of the continuity equation expressing the rate of increase of {{math|''q''}} within a volume {{math|''V''}} is: {{Equation box 1 |indent=: |equation= <math>\frac{\partial q}{\partial t} + \oint_{S}\mathbf{j} \cdot d\mathbf{S} = \Sigma</math> |cellpadding |border |border colour = #0073CF |background colour=#F5FFFA }} [[File:SurfacesWithAndWithoutBoundary.svg|right|thumb|250px|In the integral form of the continuity equation, {{math|''S''}} is any [[closed surface]] that fully encloses a volume {{math|''V''}}, like any of the surfaces on the left. {{math|''S''}} can ''not'' be a surface with boundaries, like those on the right. (Surfaces are blue, boundaries are red.)]] where * {{math|''S''}} is any imaginary [[closed surface]], that encloses a volume {{math|''V''}}, * <math>\oint_{S} d\mathbf{S}</math> denotes a [[surface integral]] over that closed surface, * {{math|''q''}} is the total amount of the quantity in the volume {{math|''V''}}, * {{math|'''j'''}} is the flux of {{math|''q''}}, * {{math|''t''}} is time, * {{math|Σ}} is the net rate that {{math|''q''}} is being generated inside the volume {{math|''V''}} per unit time. When {{math|''q''}} is being generated (i.e., when <math>\tfrac{\partial q}{\partial t}>0</math> ), the region is called a ''source'' of {{math|''q''}}, and it makes {{math|Σ}} more positive. When {{math|''q''}} is being destroyed (i.e., when <math>\tfrac{\partial q}{\partial t}<0</math>), the region is called a ''sink'' of {{math|''q''}}, and it makes {{math|Σ}} more negative. The term {{math|Σ}} is sometimes written as <math>dq/dt|_\text{gen}</math> or the total change of {{math|''q''}} from its generation or destruction inside the control volume. In a simple example, {{math|''V''}} could be a building, and {{math|''q''}} could be the number of living people in the building. The surface {{math|''S''}} would consist of the walls, doors, roof, and foundation of the building. Then the continuity equation states that the number of living people in the building (1) increases when living people enter the building (i.e., when there is an inward flux through the surface), (2) decreases when living people exit the building (i.e., when there is an outward flux through the surface), (3) increases when someone in the building gives birth to new life (i.e., when there is a positive time rate of change within the volume), and (4) decreases when someone in the building no longer lives (i.e., when there is a negative time rate of change within the volume). In conclusion, in this example there are four distinct ways that the net rate {{math|Σ}} may be altered. ===Differential form=== {{see also|Conservation law|conservation form}} By the [[divergence theorem]], a general continuity equation can also be written in a "differential form": {{Equation box 1 |indent=: |equation=<math>\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = \sigma</math> |cellpadding |border |border colour = #0073CF |background colour=#F5FFFA }} where * {{math|∇⋅}} is [[divergence]], * {{math|''ρ''}} is the density of the amount {{math|''q''}} (i.e. the quantity {{math|''q''}} per unit volume), * {{math|'''j'''}} is the flux of {{math|''q''}} (i.e. '''j''' = ρ'''v''', where '''v''' is the vector field describing the movement of the quantity {{math|''q''}}), * {{math|''t''}} is time, * {{math|''σ''}} is the generation of {{math|''q''}} per unit volume per unit time. Terms that generate {{math|''q''}} (i.e., {{math|''σ'' > 0}}) or remove {{math|''q''}} (i.e., {{math|''σ'' < 0}}) are referred to as [[sources and sinks]] respectively. This general equation may be used to derive any continuity equation, ranging from as simple as the volume continuity equation to as complicated as the [[Navier–Stokes equations]]. This equation also generalizes the [[advection equation]]. Other equations in physics, such as [[Gauss's law|Gauss's law of the electric field]] and [[Gauss's law for gravity]], have a similar mathematical form to the continuity equation, but are not usually referred to by the term "continuity equation", because {{math|'''j'''}} in those cases does not represent the flow of a real physical quantity. In the case that {{math|''q''}} is a [[Conservation law (physics)|conserved quantity]] that cannot be created or destroyed (such as [[energy]]), {{math|1=''σ'' = 0}} and the equations become: <math display="block">\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = 0</math>
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